Random Matrices

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Even for the finite temperature, there is a simple relationship between the free energy and the eigenvalues. We will discuss how one can study the fluctuations of the free energy using this relationship and results from random matrix theory. We will also discuss the distribution of the spin sampled from the Gibbs measure. Yacin Ameur Lund : On the interface between Hermitian and normal eigenvalue ensembles The weakly Hermitian regime provides a natural bridge between Hermitian and normal random matrix theories. This interface was studied in an important paper of Fyodorov, Khoruzhenko and Sommers from , and has later been the subject of a number of further investigations.

In my talk I will discuss ensembles of a similar appearance, in particular from the point of view of Ward identities. The talk concerns work in progress, and is based on discussions with Sungsoo Byun, Gernot Akemann, and others. Renjie Feng Peking : Extreme gap problems in random matrix theory The spacing between eigenvalues of classical random matrices is one of the most essential problems in random matrix theory.

We will also talk about some open problems. This is the joint work with Dongyi Wei and Gang Tian. Aron Wennman Tel Aviv : Planar orthogonal polynomials and boundary universality for determinantal Coulomb gas The main objects of this talk are the planar orthogonal polynomials associated to an exponentially varying weight in the plane.

The associated weighted reproducing kernel is the correlation kernel for two-dimensional Coulomb gas in an external field, in the special case when the inverse temperature equals two. This Coulomb gas model is well studied and well-understood in the bulk regime. The boundary behavior, however, has proven more difficult to obtain for general weights.

In this talk, I will attempt to explain how to obtain the asymptotics of the ONPs as the degree tends to infinity along with the weight parameter, and how this asymptotic information may be used to obtain boundary universality of the above random processes. Of particular interest is a flow of simple curves, which induces a foliation of a region containing the boundary into smooth simple curves. The key property of the curve family is that one and the same polynomial is, in a strong approximate sense, orthogonal to lower order polynomials with respect to a family of measures on these curves, weighted by the normal velocity of the flow.

This allows us to foliate planar orthogonality into a one-dimensional problem.

Random matrices, theory and applications (opens 2018-12222)

Ke Wang Hong Kong University of Science and Technology : Random perturbation of low-rank matrices Computing the singular values and singular vectors of a large matrix is a basic task in high dimensional data analysis with many applications in computer science and statistics. In practice, however, data is often perturbed by noise. This talk is based on joint work with Zhigang Bao and Xiucai Ding. All the eigenvalues lie on the unit circle in the complex plane and they can be regarded as a determinantal point process on the unit circle.

In this talk, we discuss a generalization of this fact and also some related topics on determinantal point processes. This talk is based on a joint work with Makoto Katori Chuo University. Seung-Yeop Lee South Florida : Asymptotics of planar orthogonal polynomials corresponding to Ginibre ensemble perturbed by finite point charges The limiting distribution of the roots of planar orthogonal polynomials, is unknown except a few cases.

Random Matrices: Theory and Practice - Lecture 2

Among the few known cases is the case when the Gaussian weight is perturbed by a single logarithmic singularity in the external potential. We extend this result to a family of external potentials with an arbitrary, finite number of singularities. This work is a joint work with Meng Yang. These dynamics are natural objects of interest and in the same time tools for sampling.

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High dimensional phenomena play an important role. Maurice Duits KTH : Periodically weighted tilings and matrix orthogonal polynomials In this talk I will discuss an approach for studying random tilings of planar domains that is particularly useful for asymptotic studies of periodically weighted tilings, such as the two periodic Aztec diamond. The backbone is a double integral formula for the correlation functions for determinantal point processes defined by products of minors of block Toeplitz matrices. In general, the integrand is expressed in terms of matrix orthogonal polynomials.

In case of infinite minors the asymptotics of these matrix orthogonal polynomials can be computed explicitly and the double integrals can be further simplified, leading to explicit formulas for models such as the two periodic Aztec diamond. Assuming the typical square-root behavior at the edge of the limiting free additive convolution, we show that the Voiculescu's law holds locally on the scale slightly above the local eigenvalue spacing at the edge. Together with our previous result in the bulk, this implies the optimal rigidity of the eigenvalues and optimal rate of convergence of Voiculescu's law.

The key element of our proof is to compensate the deterioration of the stability of the subordination equations by sharp error estimates that properly account for the local density near the edge. Johannes Heiny Ruhr-University Bochum : Assessing the dependence of high-dimensional time series via autocovariances and autocorrelations In the first part of this talk, we provide asymptotic theory for certain functions of the sample autocovariance matrices of a high-dimensional time series with infinite fourth moment. The time series exhibits linear dependence across the coordinates and through time.

Assuming that the dimension increases with the sample size, we provide theory for the eigenvectors of the sample autocovariance matrices and find explicit approximations of a simple structure, whose finite sample quality is illustrated for simulated data.

Local laws of random matrices and their applications

We also obtain the limits of the normalized eigenvalues of functions of the sample autocovariance matrices in terms of cluster Poisson point processes. In turn, we derive the distributional limits of the largest eigenvalues and functionals acting on them. If the time series has a finite fourth moment, we show that the sample correlation matrix can be approximated by its sample covariance counterpart for a wide variety of models.

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This result is very important for data analysts who use principal component analysis to detect some structure in high-dimensional time series. The optimal condition for the convergence of the empirical spectral distributions turns out to be slightly weaker than normal domain of attraction. Seong-Mi Seo KIAS : Random normal matrix models with a soft edge of the spectrum In this talk, I will introduce random normal matrix ensembles with a new type of boundary conditions. This model interpolates the free boundary case, the hard edge case, and the weakly confining case.

Adapting the asymptotic expansions of planar orthogonal polynomials due to Hedenmalm and Wennman, we obtain the scaling limit of the ensemble at the edge of the spectrum and prove edge universality when the underlying potential is radially symmetric.

Friday Gernot Akemann Bielefeld : Gegenbauer and other orthogonal planar polynomials on an ellipse: orthogonality and asymptotic results We show that Gegenbauer polynomials also called ultraspherical form a family of planar orthogonal polynomials on an ellipse, enlarging the known families of planar Hermite, Laguerre or Chebyshev polynomials. Kevin Schnelli KTH : Tracy-Widom limit for sample covariance matrices We consider the edge statistics of sample covariance matrices in the asymptotic regime when both sample size and dimensionality tend to infinity.

Philippe Sosoe Cornell : Fluctuations of the overlap in 2-spin spherical spin glasses In a series of recent papers, J.

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Baik and J. Lee have used a contour integral representation for the partition function of the 2-spin spherical Sherrington-Kirkpatrick model to analyze its fluctuations via random matrix theory, including in the low temperature phase. Journal Self-citation is defined as the number of citation from a journal citing article to articles published by the same journal.

Evolution of the number of total citation per document and external citation per document i. International Collaboration accounts for the articles that have been produced by researchers from several countries. The chart shows the ratio of a journal's documents signed by researchers from more than one country; that is including more than one country address.

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  5. Not every article in a journal is considered primary research and therefore "citable", this chart shows the ratio of a journal's articles including substantial research research articles, conference papers and reviews in three year windows vs. Ratio of a journal's items, grouped in three years windows, that have been cited at least once vs. The purpose is to have a forum in which general doubts about the processes of publication in the journal, experiences and other issues derived from the publication of papers are resolved.

    For topics on particular articles, maintain the dialogue through the usual channels with your editor. Year SJR 0. Citations per document. Year International Collaboration Documents Year Value Non-citable documents 0 Non-citable documents 1 Citable documents 0 Citable documents Documents Year Value Uncited documents 0 Uncited documents 8 Cited documents 0 Cited documents Show this widget in your own website.

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